s1306 — the R-type sector, a control census
  Q1_frame_and_gradings          {"X + T(η62, π/4)": "<1e-12", "h7_0: centraliser of two generic sl(7)_0 elements in e7 contains h7_0": "<1e-12", "h7_1: centraliser of two generic sl(7)_1 elements in e7 contains h7_1": "<1e-12", "h7_2: centraliser of two generic sl(7)_2 elements in e7 contains h7_2": "<1e-12", "h7_3: centraliser of two generic sl(7)_3 elements in e7 contains h7_3": "<1e-12", "h7_4: centraliser of two generic sl(7)_4 elements in e7 contains h7_4": "<1e-12", "h7_5: centraliser of two generic sl(7)_5 elements in e7 contains h7_5": "<1e-12", "h7_6: centraliser of two generic sl(7)_6 elements in e7 contains h7_6": "<1e-12", "h7_7: centraliser of two generic sl(7)_7 elements in e7 contains h7_7": "<1e-12", "h7ᵢ diagonal (all i), weights on 56 (last axis)": [true, {"-6": 7, "-2": 21, "2": 21, "6": 7}], "max_i distance of h7ᵢ from e7": "<1e-10", "centraliser dims in e7 of two generic sl(7)ᵢ elements, i = 0..7": [1, 1, 1, 1, 1, 1, 1, 1]}
  Q2_X_sectors_per_axis          {"axis 0 (η = +1)": {"-14": 42.0, "-2": 294.0, "2": 294.0, "14": 42.0}, "axis 1 (η = +1)": {"-14": 42.0, "-2": 294.0, "2": 294.0, "14": 42.0}, "axis 2 (η = +1)": {"-14": 42.0, "-2": 294.0, "2": 294.0, "14": 42.0}, "axis 3 (η = +1)": {"-14": 42.0, "-2": 294.0, "2": 294.0, "14": 42.0}, "axis 4 (η = +1)": {"-14": 42.0, "-2": 294.0, "2": 294.0, "14": 42.0}, "axis 5 (η = +1)": {"-14": 42.0, "-2": 294.0, "2": 294.0, "14": 42.0}, "axis 6 (η = -1)": {"-14": 42.0, "-2": 294.0, "2": 294.0, "14": 42.0}, "axis 7 (η = -1)": {"-14": 42.0, "-2": 294.0, "2": 294.0, "14": 42.0}}
  Q3_controls                    {"electric SO(8) T(I, 0)": {"-14": 84.0, "2": 588.0}, "electric SO(6,2) T(η, 0)": {"-14": 84.0, "2": 588.0}, "magnetic-only T(η, π/2)": {"-2": 588.0, "14": 84.0}}
  Q4_O_reverses_the_gradings     {"max_i |O h7ᵢ O + h7ᵢ|": "<1e-12", "CONTROL max_i |O h7ᵢ O − h7ᵢ|": 12.0, "|O·X − X|; |(O·X)₊₁₄|² vs |X₋₁₄|² (axis 7)": ["<1e-12", 42.0, 42.0]}
  Q5_omega_family_under_O        {"ω = 0·π/8": {"|O·X(ω) − X(π/2 − ω)|": "<1e-12", "|O·X(ω) − X(ω)| / |X(ω)|": "1.414e+00", "|X₋₁₄|², |X₊₁₄|² (axis 7)": [84.0, 0.0]}, "ω = 1·π/8": {"|O·X(ω) − X(π/2 − ω)|": "<1e-12", "|O·X(ω) − X(ω)| / |X(ω)|": "7.654e-01", "|X₋₁₄|², |X₊₁₄|² (axis 7)": [71.698485, 12.301515]}, "ω = 2·π/8": {"|O·X(ω) − X(π/2 − ω)|": "<1e-12", "|O·X(ω) − X(ω)| / |X(ω)|": "<1e-12", "|X₋₁₄|², |X₊₁₄|² (axis 7)": [42.0, 42.0]}, "ω = 3·π/8": {"|O·X(ω) − X(π/2 − ω)|": "<1e-12", "|O·X(ω) − X(ω)| / |X(ω)|": "7.654e-01", "|X₋₁₄|², |X₊₁₄|² (axis 7)": [12.301515, 71.698485]}, "ω = 4·π/8": {"|O·X(ω) − X(π/2 − ω)|": "<1e-12", "|O·X(ω) − X(ω)| / |X(ω)|": "1.414e+00", "|X₋₁₄|², |X₊₁₄|² (axis 7)": [0.0, 84.0]}}
  receipt   PASS filed_d1414_house_frames_npz
  receipt   PASS s1288_results_machinery_rho_T_actX
  computed  PASS Q1_X_is_minus_T_eta62_pi4_and_each_h7_i_is_a_diagonal_e7_element_with_weights_6_2_and_the_unique_GL1_commuting_with_its_sl7
  computed  PASS Q2_in_every_axis_frame_X_has_sectors_minus14_42_minus2_294_plus2_294_plus14_42
  computed  PASS Q3_the_electric_controls_occupy_minus14_and_plus2_only_so_minus14_is_the_S7_side_and_plus14_the_R_type_candidate_occupied_by_the_magnetic_control
  computed  PASS Q4_O_reverses_every_axis_GL1_so_O_invariance_forces_equal_S7_type_and_R_type_sectors
  computed  PASS Q5_O_sends_X_omega_to_X_pi2_minus_omega_so_on_the_scan_only_omega_pi4_is_O_invariant_and_the_R_type_candidate_sector_is_84_sin2_omega
  argument  ARG  reading
  argument  ARG  scope
VERDICT: in every axis-aligned GL(7) frame X has an R-type-candidate sector (+14) equal in norm to its S⁷-type sector (−14); O reverses every such grading and sends ω to π/2 − ω, so O-invariance forces ω = π/4 and the equality. Literature identification UNVERIFIED; the orbit question is not run.
RESULT_HASH c1a5fa396cca7983
s1306: 5 computed, 2 receipt, 2 argument, 0 failed
